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Wikipedia says that Telstar 1 was put into a 952 x 5933, 2.6 hour 44.8° MEO orbit but doesn't really say exactly why this particular orbit was selected.

Did it perhaps behave roughly like a Molniya orbit?

uhoh
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The orbit was chosen as a tradeoff between communication capabilities and the available performance of the launch vehicle.

Study of available boosters led to the Delta configuration of the Thor as the simplest and most reliable rocket for these purposes. Its relatively limited lifting capacity set a bound of about 180 pounds for a useful orbit. This was established as: apogee 3450 miles, perigee 575 miles, inclination to equator 45 °. The apogee is high enough to give good mutual visibility between northeastern United States and western Europe. Calculations for a working worldwide system indicate the desirability of circular orbits at 6000-8000 mile elevations; however, these were not achievable with the Delta vehicle.

Source: The Telstar Experiment, included in NASA SP-32 Volume I

Organic Marble
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    I see, not just the apogee's altitude, but its location above 45° north latitude between the two "customers". And since it was essentially a short-lived demonstration/test, no need to worry about managing precession like the Molnya orbit. – uhoh Dec 10 '21 at 21:23
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    @uhoh you'd probably enjoy a read through SP-32. – Organic Marble Dec 10 '21 at 21:24
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    Perfect! It's 5:30 AM, I got up to check for clear skies to see comet Leonard but no luck so I've got some nice, quiet reading time ahead of me this morning. – uhoh Dec 10 '21 at 21:28
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    update: Oh I see what you mean! You've linked to volume I above and volume II here and both are just chock-full of detailed discussions and photos! – uhoh Dec 10 '21 at 21:39
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    There is a longer discussion of the orbital choice in the 2nd chapter, but it boils down to what I posted. BTW, there is a volume 3 as well. https://ntrs.nasa.gov/api/citations/19640001170/downloads/19640001170.pdf – Organic Marble Dec 10 '21 at 21:42
  • Four! I, II, III, IV – uhoh Dec 10 '21 at 22:02
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    Thanks! I missed IV. It was marked out in the galley-proofers writing, so I thought it didn't exist. – Organic Marble Dec 10 '21 at 22:05
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    @uhoh it appears that Telstar 1 flew a dogleg profile "The desired higher inclination of 45° was attained by yawing the second and third stages to the south of the initially established ascent trajectory plane when the vehicle had arrived at a point where the range was clear to the south. Since the energy imparted to a spacecraft is reduced by such yawing, the final apogee or perigee or both will be reduced ." – Organic Marble Dec 11 '21 at 15:20